Inductive theorem provers often diverge. This paper describes a simple critic, a computer program which monitors the construction of inductive proofs attempting to identify diverging proof attempts. Divergence is recognized by means of a \di erence matching" procedure. The critic then proposes lemmas and generalizations which \ripple" these differences away so that the proof can go through with...

Chudnovsky et al. gave a min-max formula for the maximum number of node-disjoint non-zero A-paths in group-labeled graphs [1], which is a generalization of Mader’s theorem on node-disjoint A-paths [3]. Here we present a further generalization with a shorter proof. The main feature of Theorem 2.1 is that parity is “hidden” inside ν̂, which is given by an oracle for non-bipartite matching.

We show that the full version of the so-called ‘rural hospital theorem’ (Roth, 1986) generalizes to many-to-many matching where agents on both sides of the market have separable and substitutable preferences.

We prove Edmonds-Gallai type structure theorems for Mader’s edgeand vertex-disjoint paths including also capacitated variants, and state a conjecture generalizing Mader’s minimax theorems on path packings and Cunningham and Geelen’s path-matching theorem.

Inductive theorem provers often diverge. This paper describes a simple critic, a computer program which monitors the construction of inductive proofs attempting to identify diverging proof attempts. Divergence is recognized by means of a \diierence matching" procedure. The critic then proposes lemmas and generalizations which \ripple" these differences away so that the proof can go through with...

We prove a universality theorem for learning with random features. Our result shows that, in terms of training and generalization errors, feature model nonlinear activation function is asymptotically equivalent to surrogate linear Gaussian matching covariance matrix. This settles so-called equivalence conjecture based on which several recent papers develop their results. method proving the buil...

Ramsey’s Theorem tells us that there are exactly two minimal hereditary classes containing graphs with arbitrarily many vertices: the class of complete graphs and the class of edgeless graphs. In other words, Ramsey’s Theorem characterizes the graph vertex number in terms of minimal hereditary classes where this parameter is unbounded. In the present paper, we show that a similar Ramsey-type ch...

A graph G with a perfect matching is called saturated if G + e has more perfect matchings than G for any edge e that is not in G. Lovász gave a characterization of the saturated graphs called the cathedral theorem, with some applications to the enumeration problem of perfect matchings, and later Szigeti gave another proof. In this paper, we give a new proof with our preceding works which reveal...

We incorporate externalities into the stable matching theory of two-sided markets: We establish the existence of stable matchings provided that externalities are positive and agents’ choices satisfy substitutability, and we show that the standard insights of matching theory, such as the existence of side optimal stable matchings and the rural hospitals theorem, remain valid despite the presence...

The limited usefulness of the maximum power transfer theorem in practice is argued. Inappropriately, the utility and value of the maximum power transfer theorem are often elevated to be religious icons of electrical engineering. While the theorem appears to be useful, often in real circuits the load impedance is not set equal to the complex conjugate of the equivalent impedance of the connectin...